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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_f (g01fd)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_f (g01fd) returns the deviate associated with the given lower tail probability of the F or variance-ratio distribution with real degrees of freedom.


[result, ifail] = g01fd(p, df1, df2)
[result, ifail] = nag_stat_inv_cdf_f(p, df1, df2)


The deviate, fp, associated with the lower tail probability, p, of the F-distribution with degrees of freedom ν1 and ν2 is defined as the solution to
P F fp : ν1 ,ν2 = p = ν 1 12 ν1 ν 2 12 ν2 Γ ν1 + ν2 2 Γ ν1 2 Γ ν2 2 0 fp F 12 ν1-2 ν2 + ν1 F -12 ν1 + ν2 dF ,  
where ν1,ν2>0; 0fp<.
The value of fp is computed by means of a transformation to a beta distribution, PβBβ:a,b:
PFf:ν1,ν2=Pβ Bν1f ν1f+ν2 :ν1/2,ν2/2  
and using a call to nag_stat_inv_cdf_beta (g01fe).
For very large values of both ν1 and ν2, greater than 105, a normal approximation is used. If only one of ν1 or ν2 is greater than 105 then a χ2 approximation is used; see Abramowitz and Stegun (1972).


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth


Compulsory Input Parameters

1:     p – double scalar
p, the lower tail probability from the required F-distribution.
Constraint: 0.0p<1.0.
2:     df1 – double scalar
The degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
3:     df2 – double scalar
The degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.

Optional Input Parameters


Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_stat_inv_cdf_f (g01fd) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If on exit ifail=1, 2 or 4, then nag_stat_inv_cdf_f (g01fd) returns 0.0.

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

On entry,p<0.0,
On entry,df10.0,
W  ifail=3
The solution has not converged. The result should still be a reasonable approximation to the solution. Alternatively, nag_stat_inv_cdf_beta (g01fe) can be used with a suitable setting of the argument tol.
The value of p is too close to 0 or 1 for the value of fp to be computed. This will only occur when the large sample approximations are used.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The result should be accurate to five significant digits.

Further Comments

For higher accuracy nag_stat_inv_cdf_beta (g01fe) can be used along with the transformations given in Description.


This example reads the lower tail probabilities for several F-distributions, and calculates and prints the corresponding deviates until the end of data is reached.
function g01fd_example

fprintf('g01fd example results\n\n');

p   = [ 0.9837  0.9000   0.5342];
df1 = [10       1       20.25  ];
df2 = [25.5     1        1     ];
fp  = p;

fprintf('     p      df1     df2      f_p\n');
for j = 1:numel(p)
   [fp(j), ifail] = g01fd( ...
			   p(j), df1(j), df2(j));

fprintf('%8.3f%8.3f%8.3f%8.3f\n', [p; df1; df2; fp]);

g01fd example results

     p      df1     df2      f_p
   0.984  10.000  25.500   2.837
   0.900   1.000   1.000  39.863
   0.534  20.250   1.000   2.500

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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